Annulus Segment Calculator

Name: Annulus Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at an annulus segment. This simply is a segment of an annulus. The three-dimensional equivalent is the spherical shell cap.
Enter the radius of the outer, large circle and the inner, small circle as well as the height of the segment. Choose the number of decimal places. Then click Calculate. The height must be between R - r and 2R - r.

Formulas:

i = h - R + r

a = 2 \cdot \sqrt{h \cdot (2 R - h)}

b = 2 \cdot \sqrt{i \cdot (2 r - i)}

p = 2 \cdot [R \cdot \arccos (1 - \frac{h}{R}) + r \cdot \arccos (1 - \frac{i}{r})] + a - b

A = R^{2} \cdot \arccos (1 - \frac{h}{R}) - \sqrt{2 \cdot R \cdot h - h^{2}} \cdot (R - h) - r^{2} \cdot \arccos (1 - \frac{i}{r}) + \sqrt{2 \cdot r \cdot i - i^{2}} \cdot (r - i)

Radiuses, heights, diameters and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

In an annulus segment, the annulus is divided by a straight line so that the straight line intersects the outer and inner circles. This is different for a annulus sector, which is intersected by two straight lines radiating from the center of the circles. A semi-annulus is a special case of an annulus segment in which the straight line passes through the center. This is also a special case of a annulus sector and the only shape that is both a annulus segment and an annulus sector.
The annulus segment is axially symmetrical about the perpendicular bisector, which runs through the inner height. It has no other symmetries.
The rather complicated formulas here, compared to those for a circular segment, arise from the fact that the angles of the outer and inner segments are different, and the calculation of the angles is included in the formulas. If you want to know the angles of the two circular segments, you can enter the values for R, h, and a for the outer circle, or r, i, and b for the inner circle, into the circular segment calculator.
Also the annulus stripe is part of an annulus. Unlike the similar curved rectangle, since the two underlying circles there have the same radius and are not concentric.

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